Optional random process

A stochastic process $X = (X_t(\omega),F_t)_{t\ge0}$ that is measurable (as a mapping $(\omega,t) \mapsto X(\omega,t) = X_t(\omega)$) with respect to the optional sigma-algebra $\mathcal{O} = \mathcal{O}(\mathbf{F})$.

Comments

An optional random process is also called an adapted random process.

References

[a1]	C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. Chapt. 3, Sect. 2
[a2]	H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. Chapt. 11 (Translated from German)